# Math Help - Sketch the graph

1. ## Sketch the graph

f(x)=0 for x=0
f'(x)>0 for x<-2
f'(x)=0 for x>2
f'(x)=0 for x=-2
f'(x)<0 for -2<x<2
f"(x)<0 for x<1
f"(x)=0 for x=1
f" (x)>0 for 1<x<2

Thanks

2. Originally Posted by qbkr21

f(x)=0 for x=0
f'(x)>0 for x<-2
f'(x)=0 for x>2
f'(x)=0 for x=-2
f'(x)<0 for -2<x<2
f"(x)<0 for x<1
f"(x)=0 for x=1
f" (x)>0 for 1<x<2

Thanks
I don't know how to draw in computers so I cannot show the sketch. But I can describe it.

f(x)=0,for x=0
That means the graph passes point (0,0).

f'(x)=0 for x=-2
That means the graph is horizontal at x = -2.

f"(x)=0 for x=1
That means the concavity of the graph changes at x = 1. There is an inflection point at x=1. The graph there needs not be horizontal, though.

f'(x)>0 for x<-2
The slope of the graph is positive to the left of x = -2.
So if at x = -2, the graph is horizontal, then there is maximum point at x = -2.

f'(x)<0 for -2<x<2
From x = -2 up to x = 2, the slope is negative.

f'(x)=0 for x>2
The graph is horizontal to the right of x = 2. Horizontal up to before infinity.

f"(x)<0 for x<1
The concavity of the graph is downwards to the left of x = 1.
That means from negative infinity up to x=1, the graph is like a vertical parabola that opens downward, whose vertex is at x = -2.

f" (x)>0 for 1<x<2
The concavity of the graph is upwards between x=1 and x=2. Yet, the slope there is still negative.

---------------------
Okay, if you are still with me, the graph is like a weak or tipsy number two.
From after negative infinity up to x=1 it looks like a parabola with its highest point at x = -2.
It passes through the origin (0,0), so the highest point of the "parabola" is above the x-axis.
Then at x=1 it changes concavity--it faces upward, but it is still going down.
Then at x=2 it becomes horizontal until it reaches very close to positive infinity.

3. Hello, qbkr21!

I totally agree with ticbol's analysis . . .

Sketch the graph:

$f(x) = 0$

$\begin{array}{cccc}f'(x) > 0 & \text{for }x<-2 \\
f'(x) < 0 & \text{-}2 < x < 2 \\
f'(x) = 0 & \text{for }x = 2 \\
f'(x)=0 & \text{for }x>2\end{array}$

$\begin{array}{ccc}f"(x)<0 &\text{for }x<1 \\
f"(x)=0 & \text{for }x=1 \\
f"(x)>0 & 1

Take the statements one at a time . . .

$f(0) = 0$: the graph contains the origin.

$f'(x) > 0\text{ for }x < -2$
. . To the left of $x = -2$, the graph is rising: $\nearrow$

$f'(x) < 0\text{ for }\text{-}2 < x < 2$
. . Between -2 and +2, the graph is falling: $\searrow$

$f'(2) = 0$
. . At $x = 2$, there is a horizontal tangent: $\rightarrow$

$f'(x) = 0\text{ for }x > 2$
. . To the right of $x = 2$, the graph is a horizontal line: $\rightarrow$

$f''(x) < 0\text{ for }x < 1$
. . To the left of $x = 1$, the graph is concave down: $\cap$

$f''(1) = 0$
. . At $x = 1$, there is an inflection point.

$f''(x) > 0\text{ for }1 < x < 2$
. . From $x = 1$ to $x = 2$, the graph is concave up: $\cup$

The graph looks something like this:
Code:
                      |
|
*             |
*   :     *       |
*     :         *   |     1     2
- - - + - - - - - - * - - + - - + - - - - -
-2             |  *  :     :
|    *:     :
|     *     :
|       *   :
|           *  *  *  *
|

Evidently, there is a horizontal tangent at $x = -2$,
. . but they didn't mention it.

4. Originally Posted by Soroban
Hello, qbkr21!

I totally agree with ticbol's analysis . . .

Take the statements one at a time . . .

$f(0) = 0$: the graph contains the origin.

$f'(x) > 0\text{ for }x < -2$
. . To the left of $x = -2$, the graph is rising: $\nearrow$

$f'(x) < 0\text{ for }\text{-}2 < x < 2$
. . Between -2 and +2, the graph is falling: $\searrow$

$f'(2) = 0$
. . At $x = 2$, there is a horizontal tangent: $\rightarrow$

$f'(x) = 0\text{ for }x > 2$
. . To the right of $x = 2$, the graph is a horizontal line: $\rightarrow$

$f''(x) < 0\text{ for }x < 1$
. . To the left of $x = 1$, the graph is concave down: $\cap$

$f''(1) = 0$
. . At $x = 1$, there is an inflection point.

$f''(x) > 0\text{ for }1 < x < 2$
. . From $x = 1$ to $x = 2$, the graph is concave up: $\cup$

The graph looks something like this:
Code:
                      |
|
*             |
*   :     *       |
*     :         *   |     1     2
- - - + - - - - - - * - - + - - + - - - - -
-2             |  *  :     :
|    *:     :
|     *     :
|       *   :
|           *  *  *  *
|

Evidently, there is a horizontal tangent at $x = -2$,
. . but they didn't mention it.

Good morning, Soroban.

The 4th statement is actually, "f'(x) = 0 for x = -2". Not for x = 2. So the horizontal tangent at x = -2 is mentioned. That's why I said the vertex of the "parabola" is at x = -2.

The pesky negative sign bit you.

5. Hello, ticbol!

The pesky negative sign bit you. :)
Ouch!